2. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley CHAPTER 8: Sequences, Series, and Combinatorics 8.1 Sequences and Series 8.2 Arithmetic Sequences and Series 8.3 Geometric Sequences and Series 8.4 Mathematical Induction 8.5 Combinatorics: Permutations 8.6 Combinatorics: Combinations 8.7 The Binomial Theorem 8.8 Probability

3. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 8.1 Sequences and Series  Find terms of sequences given the nth term.  Look for a pattern in a sequence and try to determine a general term.  Convert between sigma notation and other notation for a series.  Construct the terms of a recursively defined sequence.

4. Slide 8.1 - 4 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Sequences A sequence is a function, where the domain is a set of consecutive positive integers beginning with 1. An infinite sequence is a function having for its domain the set of positive integers, {1, 2, 3, 4, 5, …}. A finite sequence is a function having for its domain a set of positive integers, {1, 2, 3, 4, 5, …, n}, for some positive integer n.

5. Slide 8.1 - 5 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Sequence Formulas In a formula, the function values are known as terms of the sequence. The first term in a sequence is denoted as a 1, the fifth term as a 5, and the nth term, or the general term, as a n.

6. Slide 8.1 - 6 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Find the first 4 terms and the 9 th term of the sequence whose general term is given by a n = 4(  2) n. Solution: We have a n = 4(  2) n, so a 1 = 4(  2) 1 =  8 a 2 = 4(  2) 2 = 16 a 3 = 4(  2) 3 =  32 a 4 = 4(  2) 4 = 64 a 9 = 4(  2) 9 =  2048

7. Slide 8.1 - 7 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Finding the General Term Example: Predict the general term of the sequence 4, 16, 64, 256, … Solution: These are the powers of 4, so the general term might be (4) n.

8. Slide 8.1 - 8 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Alternating Sequence The power (  2) n causes the sign of the terms to alternate between positive and negative, depending on whether the n is even or odd. This kind of sequence is called an alternating sequence.

9. Slide 8.1 - 9 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Sums and Series

10. Slide 8.1 - 10 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example For the sequence  1, 3,  5, 7,  9, 11,  13, … find each of the following: a) S 1 b) S 5 c) S 7 Solution: a) S 1 =  1 b) S 5 =  1 + 3 + (  5) + 7 + (  9) =  5 c) S 7 =  1 + 3 + (  5) + 7 + (  9) + 11 + (  13) =  7

11. Slide 8.1 - 11 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Sigma Notation The Greek letter  (sigma) can be used to simplify notation when the general term of a sequence is a formula. For example, the sum of the first three terms of the sequence,…,,… can be named as follows, using sigma notation, or summation notation: This is read “the sum as k goes from 1 to 3 of.” The letter k is called the index of summation. The index of summation might be a number other than 1, and a letter other than k can be used.

12. Slide 8.1 - 12 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Find and evaluate the sum. Solution: = 9 + (  27) + 81 = 6

13. Slide 8.1 - 13 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Write sigma notation for the sum 5 + 25 + 125 + … Solution: 5 + 25 + 125 + …= 5 1 + 5 2 + 5 3 + … This in an infinite series, so we use the infinity symbol  to write the sigma notation.

14. Slide 8.1 - 14 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Recursive Definitions A sequence may be defined recursively or by using a recursion formula. Such a definition lists the first term, or the first few terms, and then describes how to determine the remaining terms from the given terms.

15. Slide 8.1 - 15 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Find the first 5 terms of the sequence defined by Solution: